Downstream Processing and Bioseparation - American Chemical

Chapter 3

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Theoretical Treatment of Aqueous Two-Phase Extraction by Using Virial Expansions A Preliminary

Report

Daniel Forciniti and Carol Κ. Hall Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905 A theoretical treatment of aqueous two-phase extraction at the isoelectric point is presented. We extend the constant pressure solution theory of Hill to the prediction of the chemical potential of a species in a system containing solvent, two polymers and protein. The theory leads to an osmotic virial-type expansion and gives a fundamental interpretation of the osmotic virial coefficients in terms of forces between species. The expansion is identical to the Edmunds-Ogston-type expression only when certain assumptions are made -- one of which is that the solvent is non-interacting. The coefficients are calculated using simple excluded volume models for polymer-protein interactions and are then inserted into the expansion to predict isoelectric partition coefficients. The results are compared with trends observed experimentally for protein partition coefficients as functions of protein and polymer molecular weights.

W h e n two aqueous solutions of incompatible polymers such as polyethylene glycol (PEG), and dextran (Dx) are mixed above critical concentrations, a liquid-liquid phase separation occurs (1). Proteins or enzymes added to the resulting two-phase mixture w i l l tend to partition unequally between the phases thus a l l o w i n g for the extraction of a 1particular protein. Separations techniques based on this partitioning have come to be k n o w n as aqueous two-phase extraction (2). This technique holds great promise for the isolation of proteins because it is gentle enough for the fragile products of genetic engineering and yet robust enough to be easily adapted to large scale production. Despite the projected importance of aqueous two-phase extractive techniques for future separations technology, very little is k n o w n about the molecular basis for protein partitioning. In this paper we report preliminary work aimed at developing a comprehensive theory of protein partitioning. W e focus attention on isoelectric partitioning and use statistical mechanics to examine the fundamental basis of the so called Edmonds-Ogston expression (3) and its extension to four component systems by K i n g et al. (4). This expression, 0097-6156/90/0419-0053$06.00/0 © 1990 American Chemical Society In Downstream Processing and Bioseparation; Hamel, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


54

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which is an osmotic virial equation truncated at the second term, is well suited to the description of the properties of protein-polymer-solvent systems. This paper focuses on the relationship between the intermolecular forces and the trends observed experimentally at the isoelectric point for protein p a r t i t i o n coefficients as functions of protein a n d p o l y m e r molecular weights. These trends are that at fixed polymer concentrations (on a weight/weight basis): (a) increasing the molecular weight of the protein decreases the partition coefficient and (b) decreasing the molecular weight of one polymer increases the affinity of the protein for the phase rich i n that polymer (5). A number of other authors have developed theories w h i c h attempt to explain these trends. Brooks et aL (6) and Albertsson et aL (7) have used a Flory Huggins type theory to develop partition coefficient correlations w h i c h turn out to be similar to the Edmonds and Ogston and K i n g et aL correlation, Equation 31. The Flory H u g g i n s approach has the advantage that it is analytic and the disadvantage that proteins, although rigid, are treated as flexible polymers. A l t h o u g h these authors claim that their models predict qualitatively the trends described above, they do not consider the effect of the molecular weight dependence of the P E G and D x concentration differences between the top and bottom phases (hereafter called A P E G and ADx). Although the molecular weight dependence of A P E G and A D x is small far from the critical point, it is non negligible i n many cases (1, 8). Furthermore, as we w i l l later point out, the molecular weight dependence of A P E G and A D x can act to oppose the trends observed experimentally. Baskir et aL (9) have developed a lattice approach to treat the conformations of a polymer molecule i n the vicinity of a rigid protein molecule, which they model as a hard sphere. They find that they must include attractive protein-polymer interactions i n order to predict the trends observed experimentally. W e begin our investigation by extending the constant pressure solution theory of H i l l (10, 11) (derived by h i m for a two component system) to the prediction of the chemical potential of species i n a system containing solvent, two polymers and protein. The advantage of using the constant pressure solution theory rather than the constant v o l u m e solution theory of M c M i l l a n and Meyer (12) is that extraction experiments take place at constant pressure and are therefore more conveniently related to a theory i n which pressure is an independent variable. Furthermore, since extraction experiments are conducted by adding solute to a fixed volume of solute, it is easiest to relate to constant pressure solution theory i n w h i c h m o l a l i t y (grams solute/kilograms solvent) is the natural composition variable. The theory leads to an osmotic virial type expansion and gives a fundamental interpretation of the coefficients appearing i n this expansion in terms of forces between the species. The expansion reduces to the Edmunds-Ogston expression only when certain assumptions are made namely that the f l u i d s are incompressible and that the solvent is

In Downstream Processing and Bioseparation; Hamel, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


3. FORCINITI & HALL

Theoretical Treatment ofAqueous Two-Phase Extraction 55

non-interacting. W h i l e the first assumption is reasonable, the second assumption is clearly subject to criticism. Nevertheless given these assumptions, the coefficients are calculated using simple excluded volume models for the polymer-protein interactions (the effect of attractions w i l l be considered i n a later paper). Three models of molecular shape are considered: polymers are treated as impenetrable spheres, as impenetrable cylinders (particularly applicable to PEG) and as flexible coils; proteins are always modeled as impenetrable spheres. The osmotic virial coefficients associated w i t h these three models are inserted into the expansion to predict isoelectric protein partition coefficients. The results obtained for these three models are compared with the experimental trends described previously for protein partition coefficients as functions of protein and polymer molecular weight. The most successful of the three models is the model i n which protein and dextran are treated as impenetrable spheres and P E G is treated as an impenetrable cylinder. This model predicts the observed experimental trends w i t h protein molecular weight. N o n e of the models can totally explain the dependence of partitioning on polymer molecular weight. The regime of validity for the models depends on the relative size of the protein and the polymer; the smaller the protein, the better the correlation w i t h dextran molecular weight, the larger the protein, the better the correlation with P E G molecular weight. This investigation has enhanced our understanding of the factors w h i c h contribute to the molecular weight dependence of protein partitioning. The molecular weight dependence of the protein partition coefficient results from a competition between two terms i n the partition coefficient expansion, namely the crossed second virial coefficient and the differences between the polymer concentrations i n the top and bottom phases. While the trend i n binodal concentrations tends (in part) to favor the trends observed experimentally, the trend i n the second v i r i a l coefficient tends to oppose the experimental trends.

THEORY In 1957 H i l l introduced a binary solution theory based on an analysis of the semigrand partition function i n which the pressure P, temperature T, and number of solvent particles, N j , are held fixed (10, 11). In this section, we extend his derivation to a four-component system containing solvent (component 1), two polymers (components 2 and 3) and protein (component 4). The objective of the calculation is to derive expressions for the chemical potentials of all components. Later, by equating the chemical potentials of each component i n each phase, we w i l l determine the composition of each phase and hence the protein partition coefficient w h i c h is defined to be the ratio of protein compositions i n the top and bottom phases.


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PROCESSING AND BIOSEPARATION

The theory begins w i t h a derivation of the semigrand partition function, Γ, which is defined for a system at constant Ρ and Τ that is open with respect to components 2, 3 and 4 but not with respect to 1. The semigrand partition function is given i n terms of the canonical partition function, Q , by Ι((Τ,Ρ,Ν ,μ ,μ ,μ ) = - ι μ Λ 1

2

3

4

=


Ν

6

£

τ

e 2^ /kT ^ / k T N

2

e

e V N

3

k T

A

N ,N ,N >0 2

where

Ν N N

Δ

2

ΔΝ Ν Ν = £ 2

3

4

3

e

'

3

N 2 N 3 N 4

2

4

3

4

(1)

i the isothermal-isobaric partition function,

4

s

Q(Ni.N N .N .V.T)

P V y k T

ae

s

4

v (2) Here N i and μι are respectively the number of molecules and chemical potential of component i , and V is the v o l u m e per molecule. For convenience, we define new activities a^- Δ Ν

λ

1 0 0

.

2

ΐ οοο

_ Δ

a

'

Δ

0 1 0

λ

.

3

N^ooo

3

^ Δ

a

'

0 0 1

λ

4

N^ooo

4

(3)

in terms of the absolute activities λ{ = ^ i * The subscripts 000, 100 etc. on the A's indicate the number of molecules of species 2, 3 and 4 respectively, e.g. Δ ι ο is the isothermal isobaric partition function for a kT

L /

0

system containing one molecule of species 3. Expressing Γ i n terms of the new activities we obtain a

-= 1 +

N

*ooo

2

J £ N

4

* O N

N2

2

2

a *a * 3

! N

N

3

4

! N

N

4

!

Ν

Λ

2

Ν

3

Ν

(4)

4

where the prime on the sum indicates the restriction that 2 + N + N * 0 and

N

3

4

N +N +N -l

N ! N ! N ! Δ 000 2

3

4

L

100

2

3

4

N

ΝΓ*

+ N Τ

N, N ^010 ^001

+

N V

N N N 2

3

4

4


(5)

3. FORCINITI & HALL


Expanding the natural logarithm of Γ /Δοοο (Equation 4) and using the relationship between μι and Γ given i n Equation 1 we obtain μ ', _μ (Ρ,Τ,Ν ,Ν ,Ν )-μ (Ρ,Τ,0,0,0 [ r , l , a , a , a )= n= kT^'^" '" '" 'kT 1

t

ρ τ

)

T

2 2

3

4

3

1

2

3

4

1

4

(6)

ι

G a >a 3a
θ ιο=

1

-Νιθ ι= οοι=Νι

=>θ οι=

1

10

1

χ

0

2 ΐ

Ν

0

χ

0 0

-

10

ΐθ 00 2

=

Χ

0

200" 100 Χ

2

- 2! Νι θθ20 = 020 - ο ι ο Χ

χ

- 2! Ν θ = Χ - Ν θ =Χιιο1

1

Ν

0 0 2

= Χ

2

2

0 0 2 χ

1 1 0

ΐθιοΐ

W

10Γ

- NjOoi^Xoir

-Χοοι ιοο οιο χ

Χ

100 001

χ

οιο οοι

Χ

χ

The molarity of components 2, 3 and 4, m j , defined by H i l l to be N i / N i , can be obtained by applying the Gibbs-Duhem equation

m = a;

da

;

P.T.aj.a,

â

j,k*i

(8)

to Equation 6, thereby obtaining an expansion of the molalities of components 2 through 4 i n terms of the activities of components 2 through 4. For example m (P,T,a ,a ,a )= £ i 9 {P,T) a ^ a ^ i.j.k 2

2

3

4

ijk


(9)

58

DOWNSTREAM PROCESSING AND BIOSEPARATION

with similar expressions for m and m . U s i n g standard techniques, these series can be inverted to yield expansions for the activities a2, a and a4. These may, i n turn, be inserted into Equation 6 to yield an expansion for μ'ΐ, which to second order i n the molality is 3

4

3

- ^ r = m +m +m - 9 2

3

4

a

- Ö oo 2 - Ö m


2

m

020

m m - 0

1 1 0

2

a

3

- θ

3

0 0 2

πι

m m - 0

l o l

i2

2

4

o l l

m m 3

4

,

4

(10)

or they can be used directly to obtain expressions for μ4 μ ( Ρ , Τ , ι η ^ ' , m ' ) = μ ^ * RT (In m - 2 9 4

2

,

3

4

4

2

-2Θ

-

2

0 0 2

0 0

θ

2

θ

i2

j

2

m "2-θ 4

ι +

ö

1 0 l Ö 0 H

m

2

4

«2

*

2

θ ο ο 2

θ

1 0 1

m - 6 2

2

m -- θ

1 0 1

1 0 1 ™ 2 ^ +

m - 9

0 0 2

0 1 1 ™ 3

m

3

»2

m

0 1 1

0 1 1

ΐ

η

3

4

ι 2

m

3 )

(11)

where μ ^ = ΚΤ1ηί^οΝΐ \ Δοοι / 4

Γ

)

(12)

Changing from the simple molality of H i l l ,

to the more conventional

definition of molality, mi = N i M i / N i ( 1 0 0 0 ) , where M i is the molecular weight of species i , we find μ JP/T,m ,m ,m ) = 2

3

4

(Ρ,Τ,Ο,Ο,Ο ) -

μ ι

+ |-m2 + 2

^ ni3

2

(m + m + m 2

3

4

+1- rru + a m m + e m rru + f m 2

2

3

2

3

m4 )

and U4 (P,T,m ,m ,rru ) = μ/ * + RT (in m* + g m» + e m + f m + 0(m )) 2

6

3

2

2

3

where c/2 = - Q M 11000 200

l

N

a

d/2 = - θ

0 2 0

Μ

ι/

1 0 0 0 N

a


(14)

3. FORCINITI & HALL


g/2 = - e o o 2 i / M

1 0 0 0 N

e = -0 iM /lOOON


1 O

1

a

a= -e

1 1 0

M /1000N 1

f = - θοιχΜ^ 1 0 0 0 N

a

a

a

( 1 5 )

Equations 13 and 14 have the same functional f o r m as that postulated by Edmonds and Ogston (3) and later generalized by K i n g et aL (4). The significance of the work presented here is that it enables us to give a fundamental interpretation of the coefficients and the reference potential in terms of forces between the species. It also allows us to relate these coefficients to the virial coefficients which appear i n the M c M i l l a n - M e y e r virial expansion (12) of the osmotic pressure. In the equations of Ogston and of K i n g et aL the coefficients are set equal to the virial coefficients of the M c M i l l a n - M e y e r v i r i a l expansion, but, as we shall see, these coefficients are equivalent only when certain assumptions are made. In order to relate the coefficients a - f to the forces between molecules it is necessary to evaluate the partition function Δ Ν since the coefficients a - f are, via Equations 7 and 15, functions of the A s . Equation 5 may be rewritten i n terms of the configurational partition function ΖΝ Ν2Ν N α

Δ

3

as

4

PV/kT

Ν Ν Ν ~Σ 2

3

Z

N N^ N 1

3

e

4

3N,

N !N !Nj!N !A, 1

2

4

4

Λ

3N

2

2

Λ

3N

3

3

Λ

3N

4

(16)

4

where A j is the thermal wavelength of species i . If it is assumed that the solvent molecules do not interact with each other, i.e. Z

N N N N = V 1

2

3

4

N

L

Z

N

2

N

J

N

4

(

1

7

)

and that the solvent and solution are incompressible, i.e. V = V + N v + N3V3+ N4V4 0

2

(18)

2

where V is the total volume of the solvent and Vj is the molar volume of the solute i , then the summation i n Equation 16 contains only one term and Equation 16 becomes, Q

A

_ C 4

P

XT

N

2

+ Ν * V2

+

N

I XT

t

XT

3

V3 t

+N A

!N !N !Λ 3

4

4

3

2

V4 )/kT Q Q Z , , N

N

>

A

Λ

3N

3

3

A

3

N

N

N A

*

Λ4


,1

n\

(19)

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PROCESSING

AND BIOSEPARATION

where Q

0

Vo

=

Ai

3 N l

N l

Ni!

(20)

Here we have approximated V by V

since the solution is assumed to be

0

dilute. The Z's can be evaluated by relating them to the virial coefficients Downloaded by UNIV OF CALIFORNIA SAN DIEGO on June 5, 2015 | http://pubs.acs.org Publication Date: January 24, 1990 | doi: 10.1021/bk-1990-0419.ch003

in the M c M i l l a n and Meyer theory expansion for the osmotic pressure, Π, i n the density of each species, Pj = Ν. / V . For the case of three solutes i n a solvent, its expansion is ^

= Σ Ρ Ι + Β 2 Ο Ο p +Bo20 Ρ3 +Βοο2 p 4 2

2

2

2 + 2

B i i o p 2 p 3 + i o i P2P4+2B011 P 3 P 4 + 2 B

(21) where

B oo= - j J [exp (-w (r, μ 2

22

Τ) /kT) -l]

χ9

4 r dr 2

K

(22)

0

Β iio= - ^ J [exp (-w (r,

η / kT) -l] 4 ^ d r 2

23

0

(23)

w i t h similar expressions for the other coefficients. The wy are the potentials of mean force (or effective potential energy) between isolated molecules of type i and j i n a sea of solvent at chemical potential, μι. Using the w e l l k n o w n relationships between the Z's and the virial coefficients (11-13), û _ 4 Β »002-vf "vT v

with similar relationships for θ a

2 0 0

_ 2 , v

«no-

v

V l

+

(24)

and θ 3

B

0 2 0

, and

1 1 0

v f ^ 7

(25)

with similar relationships for θ and θ ι . The second virial coefficient By may be related to the second virial coefficients A y which are given i n terms of weight per volume units as 0 1 1

01


3. FORCINITI & HALL

Theoretical Treatment of Aqueous Two-Phase Extraction 61 A- ·=

a

R..

(26)

where N is the Avagodro's number. The coefficients are then given by a


g/2

c/2 =

M4 A44 y_4 1000NL N v i " v i Mi

d/2 =

2

a

Mi

2

N

vi " v i

a

a

2

4

2

2

I

2

3

2 3

a

3

4

a

3

2

2M M A Mi f =, 1000N [ N V!

M 2M M A 4 v v e - 1000NL Ni vy " v f v~j x

a

2M M A Mi v v a = -1000N N V i v i Vl

M 2 A22 V2

1000N;

M3 A33 V3J M 1000NL N V! " V i

4

v

3 4

v

a

3 i

v

4

(27)

Comparison of Equations 12, 13, and 27 with the Edmonds-Ogston equation and its extension by K i n g et aL indicates that Edmonds-Ogston and K i n g et aL equations are valid only when the solvent and solute are incompressible, the solvent is non-interacting and

»V:

(28)

and 2A

MjMi

- » V : + V:

(29)

Clearly the Edmonds-Ogston and K i n g et aL equations are not valid if the proteins are treated as hard spheres since i n that case A 4 4 M 4 / N = 4v4. If the proteins are treated as flexible coils (without excluded volume) then A 4 4 M 4 / N » 4v4. The validity of Equation 29 depends on the relative size of the protein and polymer. Calculations not shown here indicate that for the cases considered here, Equation 29 is generally valid. If we assume that Equations 28 and 29 are valid, that the solvent and solution are incompressible, and that the solvent is non-interacting, we can use Equation 14 to investigate the dependence of the partition coefficient on the nature of the interactions between the species. To obtain the protein partition coefficient, the protein chemical potentials given by Equation 14 are equated i n both phases. Since the reference chemical potentials are the same i n both phases, we obtain: 2

2

a


a

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PROCESSING A N D BIOSEPARATION

= g (mP - m? ) + e

In Κ» = In

πΐ

K

B

-m

2

T

) + f (π* - m Β

3

T

)

(30)

Β

Converting to weight fraction, i —IÖÖÖ~~ ^ l * * i g h * fraction of solvent, and dropping the first term on the right hand side of Equation 30 (since it is negligible compared to the others), we finally obtain


w

lnK = p

τ w ln^5

=

w

e r e

w

s

e

w e

Β

= M J ^ l n - ^ r - 2Α ΔΡΕΟ + 2A ADx 24

4

W

J4

J

]

(31) where

APEG

and Δ^χ, the driving forces for separation, are given by

Equation 31 w i l l be used to investigate the dependence of the protein partition coefficient on polymer and protein molecular weights. Aside from the trivial dependence o n M i n Equation 31, the dependence of K on molecular weight comes from its dependence o n the second virial coefficients and on the two polymer driving forces, A P E G d A D X - We w i l l f i n d theoretical and experimental grounds to believe that A 4 and A decrease as the P E G or D x molecular weight increases. A s indicated by Equation 31, this molecular weight dependence opposes the trends observed experimentally that were described i n the introduction. O n the other h a n d w e have data f r o m the literature a n d f r o m o u r o w n experiments (8) which show that A D x increases with P E G and D x molecular weights and that A P E G increases with P E G and D x molecular weights. The increase i n A P E G with P E G molecular weight and the increase i n A D x with Dx molecular weight favor the trend, but the increase i n A P E G with D x molecular weight and the increase i n A D x w i t h P E G molecular weight oppose the trend. These dependences w i l l be examined i n the next section. 4

p

a

n

2

3

4

THE DEPENDENCE OF THE SECOND VIRIAL COEFFICIENTS O N MOLECULAR WEIGHT: THREE MODELS In order to determine the dependence of the virial coefficients on polymer and protein molecular weight it is necessary to specify how the potential of



3. FORCINITI & HALL


mean force appearing i n Equations 22 and 23 depends on protein and polymer molecular weights. A l t h o u g h the forces between molecules i n aqueous two phase systems have many origins, i n this paper we consider only excluded volume forces, that is, forces w h i c h arise because two molecules cannot occupy the same space at the same time. Electrostatic interactions are neglected since we focus on the isoelectric point. Attractions are also neglected. This approach has also been used by Edmonds and Ogston (3) to model phase formation and by A t h a and Ingram (14) to model protein precipitation by P E G . While the first agrees qualitatively w i t h experiment, the second fails to predict trends w i t h molecular weight. Recently, however, Mahadevan and H a l l (15) have been able to reproduce qualitatively the trends observed experimentally for protein precipitation w i t h protein and P E G molecular weight by considering o n l y excluded volume forces. This suggests that it is worthwhile to see if excluded volume forces alone are responsible for the trends observed w i t h protein and polymer molecular weight. The excluded volume forces may be calculated on the basis of very simple models of molecular shape. By modeling proteins as rigid spheres and polymers as either rigid particles or flexible coils we hope to learn what role excluded volume effects have i n partitioning. W h i l e the modeling of polymers as r i g i d particles is somewhat questionable, the modeling of proteins which have a compact structure as rigid particles is common (9,14,16) especially for globular proteins. Three models of excluded volume forces are considered: In the first model, called the sphere-sphere model, the proteins and polymers are modeled as rigid spheres of radii, R and Rj respectively. In this case, A is given by 4

A

-

i 4

N

2 M i M [ J I C ( R + Ril

es

4

4

In the second model, the protein is modeled as a rigid sphere and the polymer is modeled as a long thin cylinder of length L . In this case A is given by i 4

Δ A = 2M M

κ R L +^ π R

N

i 4

{

4

2

4

3

A

(34)

In the third model the polymer is modeled as a flexible coil while the protein is modeled as a rigid sphere. The second virial coefficients for such a sphere-coil model have been calculated analytically by Hermans (17) who assumed that the segments of the flexible particle do not interact with each other and are Gaussian distributed. The resulting cross second virial


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AND BIOSEPARATION

coefficient for both long and short polymer (as measured by the ratio of the polymer root mean squared end to end distance, H and the protein radius, R4) are


0 /

where q(k) = (2k - l)(2k + l)(2k + 3)(2k + 4)(2k + 6). The parameters R4, Rj, L , and HQ used i n all three of the models above can be related to the molecular weight of the various species using expressions obtained from the literature. For dextran, which is a branched, flexible polymer the radius was taken to be R = 6.6 χ 10" M 0-43 9

3

(36)

3

where M is the number average molecular weight of dextran and R is i n units of centimeters (18). For P E G , which is a linear polymer with a helical configuration (19) the length of the fiber (in centimeters) was approximated (14) by 3

3

L =9.37xlO- M 1 0

(37)

2

while the end-to-end distance (in centimeters) was approximated (16) by Ho = 6.527 χ ΙΟ" M 9

2

0

(38)

5 2 6

For cases i n which P E G is modeled as a sphere, we have taken R to be the radius of an equivalent sphere, R = .38 H as prescribed by Flory (20). The values for the protein radius, R4, were taken from measurements of the h y d r o d y n a m i c r a d i i of the i n d i v i d u a l proteins. Thus for l y s o z y m e , R = 2.06 χ 10" cm. ( M = 14,100); for chymotrypsin, R = 2.25 χ 10" cm. ( M = 23,200), for albumin; R = 3.61 χ 10" cm. ( M = 65,000); and for catalase, R4 = 5.22 χ 10" c m . ( M = 250,000) (16). Substituting the values of R R , 1*4, L and H into the expressions for the cross second v i r i a l coefficient allows us to determine h o w the 2

2

7

4

0

4

4

4

7

4

7

4

4

2 /

3

0


7


3. FORCINITI & HALL


crossed virial coefficient changes as a function of molecular weight for the three models. The three models predict a nonsimple relationship between the second virial coefficient and the molecular weight of the various species which depends on the relative size of protein and polymer. The general trend is the decrease i n the second crossed virial coefficient as the size of the particles increases. This is i n agreement with scaling theories (21-23) which predict that the crossed second virial coefficient scales the same as the pure second virial coefficient w h i c h itself decreases w i t h increasing molecular weight. It is i n disagreement, however w i t h the recent experimental data of K i n g , et aL (4) w h o found that the crossed second virial coefficient decreases with increasing polymer molecular weight but increases with increasing protein molecular weight. The discrepancy has several possible explanations: the problem may be that our approximations for A are too simple or that attraction cannot be neglected, or the problem may be that the K i n g et aL measurements were conducted away from the isoelectric point possibly allowing electrostatic effects to enter the second virial coefficient measurement. i 4

THE DEPENDENCE OF THE POLYMER DRIVING FORCE POLYMER MOLECULAR WEIGHT

ON

The dependence of the separation d r i v i n g forces, A P E G d Δρχ, on polymer molecular weight is well documented i n the literature (1). For example, Figure 1 shows schematically h o w changes i n the molecular weight of one of the polymers lead to shifts i n the binodal. In the figure, the lower curve is for a higher molecular weight P E G than is the upper curve. The tie lines for the two binodal curves are roughly parallel. It can be seen from the figure that if the polymer concentration o n a weight by weight basis is held fixed (say at Point A ) , then increasing the P E G molecular weight (at fixed D x molecular weight) w i l l result i n an increase in A P E G and A . Similarly, reference to binodals available i n the literature (1) shows that increasing the D x molecular weight (at fixed P E G molecular weight) w i l l also increase the values of A P E G d A Q X but not as much as for the P E G molecular weight increase. Clearly the increases i n A P E G d A w i l l be greater the closer the tie line is to the critical point. Equation 31 shows that increases i n the P E G or D x molecular weights act to increase the strength of two competing terms i n the protein partition coefficient. a

D

n

X

a

n

a

D

n

X

DEPENDENCE OF THE PROTEIN PARTITION COEFFICIENT POLYMER AND PROTEIN MOLECULAR WEIGHTS

ON

The partition coefficients predicted by the theory for the four globular protein, lysozyme, chymotrypsin, albumin and catalase were determined by inserting into Equation 31 the second virial coefficients calculated for each


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of the three models considered, a n d the experimental values for the driving forces, A P E G and Δ ^ χ . Kp vs PEG molecular weight. The best results are obtained for the sphere-cylinder model i n which the protein and dextran are modeled as spheres and the P E G is modeled as a cylinder. The sphere-sphere model yields mixed results while the sphere-coil model yields trends opposite to those observed experimentally. W e have considered the case i n w h i c h both p o l y m e r s are m o d e l e d as flexible coils a n d f o u n d that the experimental trends are not predicted. Figure 2 shows the partition coefficient versus P E G molecular weight predicted by the sphere-cylinder model for the four proteins when the dextran molecular weight is fixed at 23,000 a n d the m i x t u r e composition is P E G : 6%; Dx: 8%. Thus for l o w values of the D x molecular weight, the model predicts the observed experimental trend, i.e. that the partition coefficient decreases with increasing P E G molecular weight. Figure 3 shows the partition coefficient versus P E G molecular weight predicted b y the sphere-cylinder model for the four proteins when the dextran molecular weight is increased to 180,000. In this case the experimental trend is predicted only for P E G molecular weights below 10,000. Kp versus Dx Molecular weight. Modeling D x as a sphere and P E G as a sphere or a cylinder gives the right trend as a function of D x molecular weight for D x molecular weights greater than 150,000. This trend is that K increases as D x molecular weight increases. A t lower values of the D x molecular weight these models predict the experimental trend, only for the smallest proteins, chymotrypsin and lysozyme. This is illustrated i n Figure 4 w h i c h shows the partition coefficient versus D x molecular weight predicted b y the sphere cylinder model for the four proteins when the P E G molecular weight is fixed at 7500 and the mixture composition is P E G : 6%; Dx: 8%. Keeping D x modeled as a sphere but considering P E G to be a flexible coil increases the protein molecular weight below w h i c h the experimental trend is predicted. W e have also considered the case i n which both polymers are taken to be flexible coils and found that the theory fails to explain the experimental trend. p

Kp versus Protein Molecular Weight. Modeling D x as a sphere and P E G as a cylinder gives the right trend w i t h protein molecular weight namely that K p decreases as protein molecular weight increases for a l l values of P E G and D x molecular weights. See Figures 2, 3 and 4. The sphere-sphere and sphere-flexible coil models predict the experimental trend except for l o w values of the dextran molecular weight.




3. FORCINITI & H A L L

% D X (Wt./Wt.)

Figure 1. Schematic showing binodal curves for two systems at the same Dx molecular weight but different P E G molecular weights. The lower curve is for the higher P E G molecular weight. A tie line is shown through point A .

c ω ο Φ

s Ν

»•—

ο ϋ

j Lysozyme

oh

-5.0

-

Chymotrypsin

^ Albumin

C

ο

r

S.

-10.0 ^ ^ ^ ^ ^ ^

Ο)

/Catalase

ο -15.0 3000.0

ι

I 8000.0

I

.

I

1.3Ε+04

. 1.8Ε+04

PEG Molecular Weight Figure 2. Predicted protein partition coefficient versus P E G molecular weight for lysozyme, chymotrypsin, a l b u m i n a n d catalase. Dextran molecular weight is 23,000 and polymer composition is P E G : 6%; Dx: 8%.


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\ Lysozyme^Chymotrypsin C


ω ο Ε ω ο ϋ c g

Albumin

-10.0

Έ

CO Û. Ο)

-15.0 Catalase

ο -20.0 3000.0

1.3Ε+04

2.3Ε+04

3.3Ε+04

PEG Molecular Weight Figure 3. Predicted protein partition coefficient versus P E G molecular weight for lysozyme, chymotrypsin, a l b u m i n and catalase. Dextran molecular weight is 180,000 and polymer composition is P E G : 6%; Dx: 8%.

j Lysozyme

c ω 'δ Ε ω ο ϋ c ο

Ν

Chymotrypsin

-5.0 Albumin

-10.0

CO CL Ο)

9

^

Ν

Catalase

-20.0 -25.0 ' 2.3E+04 1

1

1

7.3E+04

« ' 1.2E+05

1 « 1.7E+05

1 ' 2.2E+05

1 1 2.7E+05

Dextran Molecular Weight Figure 4. Predicted protein partition coefficient versus D x molecular weight for lysozyme, chymotrypsin, albumin and catalase. P E G molecular weight is 7,500 and polymer composition is P E G : 6%; Dx: 8%.


3. FORCINITI & HALL



CONCLUSION AND DISCUSSION In the paper we have derived an expression for the protein partition coefficient w h i c h can be used to understand the molecular basis of partitioning. By playing with the equation we can learn what effect each type of intermolecular force (and interspecies force) can be expected to have on the partition coefficient. By working backwards from measured values of the partition coefficients we can learn something about the forces between proteins and polymers i n solution. A s a result of this work we have learned that excluded volume forces alone are not sufficient to predict the trends observed experimentally for protein coefficients as a function of molecular weight. This was a surprise to us since models of PEG-induced protein precipitation based on excluded volume forces only have been quite successful (15). It appears that attractions between species play a strong role i n partitioning and should be included i n any modeling effort. This is i n agreement with the conclusions reached by Baskir, et aL (9) w h o found it necessary to include an attractive term i n their lattice theory of aqueous two-phase extraction i n order to obtain reasonable values for the free energies. The inclusion of an attractive term might also improve the agreement between theoretical and experimental predictions of the cross second virial coefficient. One might also question whether the truncation of the expansion at the second v i r a l coefficient level and the neglect of the protein-protein interaction term i n Equation 30 is valid (24). While the inclusion of threeb o d y terms (and of the protein-protein term) s h o u l d i m p r o v e the comparison w i t h experiment quantitatively, we suspect that the trends predicted for the protein partition coefficient as a function of molecular weight w o u l d be the same. W e have also shown that the Edmonds-Ogston expression and its extension by K i n g , et aL are valid only if one assumes that the fluids are incompressible, that the solvent is non-interacting, and that Equations 28 and 29 are valid. The incompressibility assumption seems reasonable, but the lack of interaction between solvent and solutes seems less reasonable. We are currently investigating the consequences to the theory if this assumption is dropped. The assumption of the validity of Equations 28 and 29 is not a problem since these equations are valid when both species are flexible coils; they are also valid i n the case when one species is rigid and the other is a flexible coil. One of the problems that we have encountered i n this work is that experimental data i n the literature on A P E G d A , while quite extensive, is not extensive enough (nor accurate enough) for us to thoroughly examine the trends predicted by these models under a variety of conditions. O u r current and future w o r k therefore includes some experimental measurements of binodals (and partition coefficients) and some modeling work to obtain equations for estimating A P E G and Δ^χ (8). a

n

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X


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ACKNOWLEDGMENTS This work was supported by the National Institutes of Health (Grant # 1 ROI GM40023-01), the N a t i o n a l Science Foundation (Grant # C B T 8720284) and the N o r t h Carolina Biotechnology Center.


LITERATURE CITED 1. Albertsson, P. Α., Partition of Cell Particles and Macromolecules; J. Wiley & Sons, New York, 1986. 2. Walter, H.; Brooks, D. E.; Fisher, D., Partition in Aqueous Two-Phase Systems; Academic Press; Florida, 1985. 3. Edmonds, E.; Ogston, A. G., Biochem.J.,1968, 109, 569. 4. King, R. S.; Blanch, H. W.; Prausnitz, J. M., AIChEJ.,1988, 34, 1585. 5. Johansson, G. In Partition in Aqueous Two Phase Systems, Walter, H.; Brooks, D. E.; Fisher, D., Eds., Academic Press; Florida, 1985; pp. 161-219. 6. Brooks, D. E.; Sharp, Κ. Α.; Fisher, D., in Ref. 2. 7. Albertsson, P. Α.; Cajarville, Α.; Brooks, D. E.; Tjerneld, F., Biochem. Biophys. Acta., 1987, 926. 8. Forciniti, D.; Hall, C. K.; Kula, M. R., to be published. 9. Baskir, J. N.; Hatton, Τ. Α.; Suter, U. W., Macromolecules, 1987, 20, 1300.5 10. Hill, T. L.,J.Am. Chem. Soc. 1957, 79, 4885. 11. Hill, T. L.,J.Chem. Phys.. 1959, 30, 93. 12. McMillan, W. G.; Meyer, J. Ε.,J.Chem.Phys.,1945, 13, 276. 13. McQuarrie, D. M., Statistical Mechanics, Harper and Row, New York, 1976. 14. Atha, D. H.; Ingram, K. C.,J.Biolog. Chem., 1981, 256, 12108. 15. Mahadevan, H.; Hall, C. K., to be published. 16. Tanford, C., Physical Chemistry of Macromolecules, Wiley and Sons, New York, 1963. 17. Hermans.J.,J.Chem.Phys.,1982,77,2183. 18. Senti, F. R.; Hellman, Ν. N.; Ludwing, Ν. Η.; Babcock, G. Ε.; Tobin, R.; Glass, C. Α.; Lamberts, B., T. Poly.Sci.,1955, 27, 527. 19. Koenig, J. L.; Angood, A. C.; T. Poly.Sci.,A-2, 1970, 8, 1797. 20. Flory, P. J., Principles of Polymer Chemistry. Cornell University, Ithaca, New York, 1953. 21. Joanny, J. F.; Liebler, L.; Ball, R.,J.Chem. Phys., 1984, 81, 4640. 22. Broseta, D.; Liebler, L.; Joanny, J. F., Macromolecules, 1987, 20, 1935. 23. Kosmas, M. K.; Freed, K. F.,J.Chem.Phys.,1978, 69, 3647. 24. Haynes, C.; Prausnitz, J. M., (private communication), 1989. RECEIVED September 28, 1989


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